Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/234

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210
DIFFERENTIAL CALCULUS


Transposing one term in each to the second member and dividing, we get

.

Therefore, from (A), and ,
giving and .
Substituting these values in (B), we get the envelope

,

a pair of conjugate rectangular hyperbolas (see last figure).


EXAMPLES

1. Find the envelope of the family of straight lines , being the variable parameter.

Ans. , ; or .[1]

2. Find the envelope of the family of parabolas , being the variable parameter.

Ans. , ; or .

3. Find the envelope of the family of circles , being the variable parameter.

Ans.

4. Find the equation of the curve having as tangents the family of straight lines , the slope being the variable parameter.

Ans. The ellipse .

5. Find the envelope of the family of circles whose diameters are double ordinates of the parabola .

Ans. The parabola .

6. Find the envelope of the family of circles whose diameters are double ordinates of the ellipse .

Ans. The ellipse .

7. A circle moves with its center on the parabola , and its circumference passes through the vertex of the parabola. Find the equation of the envelope of the circles.

Ans. The [cissoid] .

8. Find the curve whose tangents are , the slope being supposed to vary.

Ans. .

9 Find the evolute of the ellipse , taking the equation of normal in the form , the eccentric angle being the parameter.

Ans. , ; or .

10. Find the evolute of the hypocycloid , the equation of whose normal is , being the parameter.

Ans. .


  1. When two answers are given, the first is in parametric form and the second in rectangular form.